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The Numbers Game

I try to build on the work of others when I write, so by way of introduction to this topic, I'd suggest you go back and take a second look at the recent Mark Young article, "Devil's in the Details."

Mark's article discusses various probabilities associated with when you can expect to draw a certain card in your deck, depending on how many copies of that card you are playing. Specifically he takes a look at the idea of "just how important is one slot?" and ultimately concludes that a difference in one slot is surprisingly significant the average sixty-card deck.

Today I'm going to delve into the practical application of this knowledge by discussing the fundamentals of how to decide on the number of slots that should be devoted to a given card in a sixty-card deck.

The first task is to determine how many slots you have to work with.

Generally speaking, 18-26 slots must be devoted to mana producers, but the topic of mana balancing is complex enough to warrant an article all its own. From here on out I'll assume, for simplicity's sake, that your deck's mana base is already complete and that you have already decided which cards are going to make the cut in your deck. The only remaining task is to figure out how many copies of these remaining cards to play.

Two situations come immediately to mind in which the number of slots you will devote to a card is inflexible.

If the card is in there specifically to be tutored up, but you rarely (if ever) want to actually topdeck it, then there is an excellent chance that you will be forced into playing exactly one copy. Look at Hana Kami or Soulless Revival in this year's Gifts Ungiven decks, for example - you are almost never happy to topdeck either one of these cards, but you still play one of each in order to make Gifts Ungiven a more powerful tutor. Ideally you would play less than one copy in order to topdeck them even less frequently, but since you must obviously play at least one copy of a card if you are to be able to tutor for it, you are stuck with devoting exactly one slot to it instead.

It also sometimes happens that you want to play as many copies of a given support card as possible, but the rest of your deck is already too locked-in-place to make room for the number you desire. Decks like U/G Madness, for example, require such a rigid skeleton to succeed on a basic level (4 Wild Mongrel, 4 Arrogant Wurm, 4 Circular Logic, and so on) that sometimes you just can't fit as many copies of a supporting card in as you would like to.

Luckily, these two scenarios are special cases. For the most part, you'll have room for 1-4 copies of any given card in your deck, and can figure out the optimal number to play at your own discretion.

There are three basic criteria for deciding how many copies of a card should be played in your deck.

1) How early do I want to draw it?
2) How good or bad is it if I draw multiple copies?
3) The answers to #1 and #2 are at odds, which is more important?

Depending on your answer to Question #1 ("How early do I want to draw it?"), you submit a vote for a certain number of copies.

Some vague guidelines (guidelines, not rules!) for how the answers to these questions can translate into votes are as follows.

If I want to play the card as early as possible (turns 1-4), that's a vote for four copies.

If I want to play the card in the midgame (turns 5-9), that's a vote for three copies.

If I want to play the card in the late game (turns 10-14), that's a vote for two copies.

If I want to play the card only in very long games (turns 15 and later), that's a vote for one copy.

Next, take a look at Question #2 ("How good or bad is it if I draw multiple copies?").

If the second copy I draw is either comparably good or better than the first copy, that's a vote for four copies.

If the second copy I draw is expected to be significantly worse than the first copy, that's a vote for three copies.

If the second copy I draw is unlikely to ever be played, that's a vote for two copies.

If the second copy I draw is outright worthless or counterproductive under almost every situation, that's a vote for one copy.

Since the answers to Question #1 and Question #2 often result in votes for different slot counts, Question #3 ("If the answers to #1 and #2 are at odds, which is more important?") is used to settle the dispute by skewing the final slot count towards the vote of whichever question is more relevant to your deck's success.

1) How early do I want to draw it?In my opening hand! I want to play that sucker as early as possible-on turn 1or turn 2 if possible - because the longer I wait, the more chances my opponent has to play the card I want to strip out of his hand.

Since I want to see Therapy in my opening hand every game, I should maximize my chances of drawing it early by playing as many copies as possible.

According to Question #1 ("How early do I want to draw it?"), the conclusion so far is that Cabal Therapy should be a four-of.

2) How good or bad is it if I draw multiple copies?
Multiple copies are fine! Each Therapy I draw will almost always still be a one-for-one, and sometimes even better than that because the previous Therapy revealed that my opponent is holding duplicates of a card I would not normally have thought to name.

According to question #2 ("How good or bad is it if I draw multiple copies?"), I have another vote for playing four copies of Cabal Therapy.

3) If the answers to #1 and #2 are at odds, which is more important?
Since they are not at odds, the conclusion is simple: play four copies!

That one was basically a no-brainer, though. Let's look at a slightly more complex example: Antonino DeRosa's winning Mono-Blue Urzatron deck from U.S. Nationals.

1) How early do I want to draw it?
Not early at all. In fact, I'd prefer most if it just hung out in my deck until after I've finished assembling the Urzatron. Unless I'm a miser and just draw the whole Tron by turn 4 or 5, it's going to take me a solid amount of time to set it up, and until I've completed it, Memnarch is just going to sit in my hand and spectate.

Since I don't want to see Memnarch until somewhere between the midgame (turns 5-9) and the late game (turns 10+), I'd say I want 2 or 3 copies.

2) How good or bad is it if I draw multiple copies?
Not so good. A second Memnarch is unplayable, so the only time I ever want a second one is if my first one gets killed. However, between Aether Spellbombs, Mindslavers, and countermagic, I have numerous ways of seeing to it that my first copy doesn't get killed, so a second copy is almost never desirable.

Since it's unlikely I'll ever play a second copy of Memnarch if I draw it, I want 2 copies.

3) If the answers to #1 and #2 are at odds, which is more important?
Another easy one. It's pretty straightforward to take a look at "I want 2-3 copies" and "I want 2 copies" and settle on exactly two copies. Two copies happens to be exactly what Antonino played.

1) How early do I want to draw it?
Just like with Therapy, I want this one as early as possible. The primary function of Seething Song in my deck is to power out Arc-Slogger on turn 3, and the Song doesn't help me "power out" Arc-Slogger any faster once I've got enough land in play to just hardcast him.

Even when I don't draw Arc-Slogger at all, Seething Song is still best early. It lets me make two plays on turn 3, like laying a Hearth Kamiand a Zo-Zu, where ordinarily I could have only played one or the other. Again, the Song doesn't help out very much with this once I've already got the enough land in play to just pay for whatever I want to play by tapping land.

Since I want it in my opening hand every time, Question #1 once again indicates I should play the maximum four copies of Seething Song.

2) How good or bad is it if I draw multiple copies?
Pretty bad, unless I am also playing late-game synergy cards like Fireball or Furnace Whelp to sink the mana into. (Since most Red decks last season were not doing this, let's assume the deck I'm building has no such late-game sinks.)

Since nothing in my deck can really make use of the mana boost once I've already played one Seething Song, I really don't ever want to see a second one.

Question #2 indicates that I should play two copies of Seething Song because multiple copies are usually outright dead draws.

3) If the answers to #1 and #2 are at odds, which is more important?
They are definitely at odds this time, so now we have to dig a little deeper.

Since I've already determined that Seething Song should go in my deck, that must mean that turn 3 Arc-Slogger is important to me. Once I have committed to a deck strategy that revolves around a quick and explosive early game, I have to come to grips with the fact that the benefits of drawing Seething Song early outweigh the downsides of drawing multiple copies late. I don't even want there to be a late game, so as long as drawing one Seething Song early lets me play the extremely broken turn 3 Slogger and put me in a position to end the game quickly and decisively, I can live with the fact that the second one might just sit in my hand and do nothing.

In this case, Question #3 settles the disagreement between the first two questions by concluding that the answer to Question #1 is more important than the answer to Question #2 is for this particular deck, and I should play four copies of Seething Song in spite of how bad drawing multiple copies can be.

Okay, one final example. Last one.

Let's look at Neil Reeves's Mono-Blue Control list from the same Nationals Top 8.

1) How early do I want to draw it?
Boomerang in this deck is a contingency plan. If a permanent resolves that Mono-Blue didn't want to hit play, Boomerang is almost always a sure-fire way to get rid of it. Since problem permanents like Pithing Needle, Genju of the Spires, and Slith Firewalker can hit play before they can be countered, I want a Boomerang somewhere in my hand as early as possible to deal with these problem cards.

Just like with Cabal Therapy and Seething Song, question #1 indicates that I should play four Boomerangs.

2) How good or bad is it if I draw multiple copies?
Real bad. In a mono-Blue control deck, card advantage is integral to my path to victory. Neil played scads of card advantage generators - four Jushi Apprentice, four Vedalken Shackles, and two each of Oblivion Stone and Thirst for Knowledge. The rest of his deck was all one-for-ones, except for Boomerang. Boomerang was the only card in his entire list that generated card disadvantage.

He did this because card disadvantage is really quite bad news for mono-Blue, as mono-Blue's preference is that it uses its countermagic to trade one-for-one with all the opponent's threats while having extra cards left over like Spire Golem and Meloku to finish the game. Although having Boomerang as a contingency plan is important, drawing two is really bad news.

Question #2 indicates that I should play two or at most three copies of Boomerang in order to avoid drawing multiples.

3) If the answers to #1 and #2 are at odds, which is more important?
Here we have about a tie in the question relevance department; it is simultaneously quite important that I draw a Boomerang early and that I do not draw more than one copy over the course of a game.

In an aggro-heavy metagame where cheap threats are likely to slip through, it might be more important to have one somehow (meaning I should go for the full four copies), but in a more Tooth and Nail and Mono-Blue Urzatron-heavy field, I'd probably only want two.

Since the field at Nationals was pretty evenly spread out, Neil apparently went for the even compromise of three copies.

So there you have it: the basics. This certainly isn't all there is to slot selection, but it is at least a good starting point. For the math whizzes in the house, I'll close with a reference table of the probabilities associated with various card slot counts.

I ran these numbers assuming you are going into turn 1 with exactly seven cards in hand, meaning you either went first and kept your opening hand or drew first and mulliganed once. If this is your game state, these are your chances of drawing...

...At least one copy in opening seven:

4-of: 40% of the time
3-of: 32% of the time
2-of: 22% of the time
1-of: 12% of the time

...At least one copy by turn 5:

4-of: 60% of the time
3-of: 49% of the time
2-of: 36% of the time
1-of: 20% of the time

...At least one copy by turn 10:

4-of: 75% of the time
3-of: 64% of the time
2-of: 49% of the time
1-of: 28% of the time

...At least two copies in opening seven:

4-of: 6% of the time
3-of: 3% of the time
2-of: 1% of the time

...At least two copies by turn 5:

4-of: 18% of the time
3-of: 10% of the time
2-of: 4% of the time

...At least two copies by turn 10:

4-of: 32% of the time
3-of: 19% of the time
2-of: 8% of the time

Bonus Section: Magical Musicians
I had a friend in high school named Avi Vinocur. His weapon of choice was once upon a time sixty-card decks - he took home 3rd at the JSS on the Queen Mary in 1998 - but he has since gone on to pursue his (considerable) talent with six-string guitars instead. It's pretty rare and more than a little exciting to me when former Magic players go on to be successful musicians, so check out his website (www.avivinocur.com) for a happier end to the story of a tournament Magic player than the usual story of Vs. or Poker.